Secondary 2 The Area and Volume of Solids

17 of the 18 questions

Each question targets one named concept from the sheet. Read them here, or print the PDF — it has working space under each one. 17 of the 18 questions are printed below. The other 1 is built on a diagram or a table of values that does not translate to the page, so it is in the free PDF — marked below where it would have come.

1. Q1Constructing the Net of Prisms

   A chocolate bar is packaged in a right prism whose base is a triangle with sides $5\text{cm}$, $5\text{cm}$ and $8\text{cm}$. The prism is $20\text{cm}$ long.

   1. List every piece that appears in the net of this package, giving the dimensions of each rectangle.
   2. How many faces does the package have in total?
   3. The three rectangles are drawn side by side as a single strip. What are the dimensions of that strip?
2. Q2Constructing the Nets of Pyramids

   A paper lantern is a regular pyramid with a square base of side $10\text{cm}$. Each of its four triangular faces has a slant height of $12\text{cm}$, measured from the middle of a base edge up to the apex.

   1. List the pieces of the net and give the dimensions of each one.
   2. Find the area of paper used, including the base.
   3. Which measurement of the pyramid appears in the net — the height of the pyramid or the slant height of a face? Explain.
3. Q3Constructing the Nets of Solids

   A packaging workshop keeps one net on file for each shape it sells. For each solid below, say how many pieces its net contains and what shape each piece is.

   1. A cube.
   2. A right prism with a regular hexagonal base.
   3. A cylinder.
   4. A pyramid with a triangular base.
4. Q4Decomposable and Truncated Solids

   A wooden toy train engine is made of two rectangular prisms glued together: the body measures $18\text{cm}\times 8\text{cm}\times 6\text{cm}$, and the cab, which measures $8\text{cm}\times 8\text{cm}\times 7\text{cm}$, sits on top of one end of the body.

   1. Find the volume of wood in the engine.
   2. The wood has a mass of $0.6\text{g}$ per cubic centimetre. Find the mass of the engine.
5. Q5Finding the Area of a Solid Using Algebra

   A locker organiser is a right prism with a square base of side $x$ centimetres and a height of $9\text{cm}$.

   1. Write a simplified expression for the total area of the prism in terms of $x$.
   2. Use your expression to find the total area when $x=5\text{cm}$.
   3. Check your answer to b) by computing the area face by face.
6. Q6Prisms

   A bird feeder is a right prism whose base is a trapezoid with parallel sides of $10\text{cm}$ and $6\text{cm}$ and a height of $4\text{cm}$. The prism is $15\text{cm}$ long.

   1. Name the solid precisely and say how many faces, vertices and edges it has.
   2. Find the area of the base.
   3. Find the volume of seed the feeder holds when it is full.
7. Q7Pyramids

   The roof of a garden gazebo is a regular pyramid with a regular octagonal base.

   1. How many faces, vertices and edges does it have?
   2. Verify Euler's relation $F+V-E=2$ for this solid.
   3. Which of its edges are congruent to one another, and which are not?
8. Q8Solids

   Here are five solids: a cube, a cylinder, a pyramid with a triangular base, a sphere and a right prism with a hexagonal base.

   1. Sort them into polyhedra and solids that are not polyhedra.
   2. State, in your own words, what makes a solid a polyhedron.
   3. Explain why a solid that has a curved surface can never be a polyhedron.
9. Q9Solids with Curved Surfaces

   This question is built around a diagram or a table of values. Open it in the PDF.

10. Q10Tetrahedrons

    A puzzle piece is a regular tetrahedron with an edge of $9\text{cm}$.

    1. How many faces, vertices and edges does it have?
    2. A tetrahedron is a pyramid. What shape is its base, and what makes it different from the other pyramids you have met?
    3. One of its faces has an area of $35{\text{cm}}^2$, to the nearest square centimetre. Find the total area of the puzzle piece.
11. Q11The Area and the Volume of Solids

    For each quantity below, say whether it is a length, an area or a volume, and give a sensible unit for it.

    1. The paint needed to cover the four walls of a bedroom.
    2. The strip of trim to be nailed around a window frame.
    3. The water a rectangular aquarium holds.
    4. The wrapping paper needed for a gift box.
    5. The distance all the way around the base of a cylindrical drum.
12. Q12The Area of Cylinders

    A cylindrical soup can has a radius of $5\text{cm}$ and a height of $12\text{cm}$. Give each answer to the nearest tenth of a square centimetre.

    1. Find the area of the label that wraps around the can (the lateral area).
    2. Find the combined area of the two circular ends.
    3. Find the total area of metal in the can.
13. Q13The Area of Decomposable Solids

    A display podium for a science fair is made of two rectangular prisms. The lower block measures $80\text{cm}\times 40\text{cm}\times 20\text{cm}$. The upper block measures $40\text{cm}\times 40\text{cm}\times 20\text{cm}$ and sits squarely on one end of the lower block's top face. Every visible outside surface is to be varnished, including the underside.

    1. Find the total area of the two blocks taken separately.
    2. Explain which parts of that total are not varnished, and find the area to be varnished.
14. Q14The Area of Prisms

    A doorstop is a right prism whose base is a right triangle with legs of $5\text{cm}$ and $12\text{cm}$ and a hypotenuse of $13\text{cm}$. The prism is $20\text{cm}$ long.

    1. Find the area of one triangular base.
    2. Find the lateral area, using the perimeter of the base.
    3. Find the total area, and also the volume of the doorstop.
15. Q15The Area of Pyramids

    The roof of a birdhouse is a regular pyramid with a square base of side $14\text{cm}$. Each triangular face has a slant height of $24\text{cm}$.

    1. Find the area of one triangular face.
    2. Find the lateral area of the roof.
    3. Find the total area of the pyramid, base included.
16. Q16The Area of Truncated Solids

    A wooden pen holder starts as a right prism with a rectangular base $6\text{cm}$ by $4\text{cm}$, then its top is sawn off on a slant. Two of its vertical edges are $10\text{cm}$ long and the other two are $7\text{cm}$ long. The slanted top face is a rectangle measuring $4\text{cm}$ by $6.7\text{cm}$ (to the nearest tenth).

    1. Two of the side faces are trapezoids. Find the area of one of them.
    2. Find the areas of the other two side faces and of the base.
    3. Find the total area of the pen holder, to the nearest tenth of a square centimetre.
17. Q17The Area of a Cube
    1. A storage cube has an edge of $7\text{cm}$. Find its total area.
    2. A second cube has a total area of $150{\text{cm}}^2$. Find the length of its edge.
    3. How much greater is the total area of the first cube than that of the second?
18. Q18Synthesis — drawing on several sheets in this topic

    A storage bin is a right prism with a rectangular base $60\text{cm}$ by $40\text{cm}$ and a height of $50\text{cm}$. A cylindrical channel of radius $10\text{cm}$ is bored straight through it from the top face to the bottom face, so that cables can pass through. Give each answer to the nearest tenth.

    1. Find the volume of material left in the bin.
    2. Find the lateral area of the outside of the prism, and the combined area of the top and bottom faces once the two circular holes are taken out.
    3. Find the total area of the finished bin, remembering the wall of the channel.

The 17 challenge problems for this topic are a separate, paid sheet and are not reproduced here.

How to do every concept on this sheet

This is the part worksheet sites usually leave out. Below is the actual reasoning behind each group of questions — not a full solution set, the decisions that get you to one. Read it before you start, or after you get stuck.

How the set is sequenced, and why that order helps

The questions are not interchangeable. Each block assumes what the block before it established, which is what lets a long topic stay memorisable: there are only a few ideas here, applied to more and more solids.

The net is the method, not a picture

Almost every area question on this sheet is answered the same way: unfold, name the pieces, add their areas. What the net makes visible is why the lateral area of a prism or a cylinder is one single rectangle rather than a stack of separate faces.

Q1 builds that rectangle by hand — three faces drawn side by side, a strip whose width is 5 + 5 + 8 = 18 cm, the base's perimeter, and whose height is the prism's own length. Q14 then uses the shortcut deliberately, and Q12 does the same thing for a can, where the label unrolls into a rectangle of width 2πr. Adding faces one at a time gives the same answer; the perimeter form is faster and much harder to lose a face in.

The slant height is not the height (Q2, Q15)

This is the single most common mistake in the pyramid questions, and it is worth settling once. The height of a pyramid runs from the apex straight down to the centre of the base. The slant height runs from the apex down the middle of a triangular face to the middle of a base edge. They are different lengths, and only the slant height appears in the area of a face.

Q2 asks the question outright — which of the two measurements shows up in the net? — and the answer is the slant height, because the net contains the actual triangular faces and the pyramid's height is nowhere in it. It is inside the solid, not on its surface.

Decomposable solids: what disappears when you glue two pieces together (Q13)

Volume and area behave completely differently here, and the difference is the whole point of the block. Q4 glues a cab onto a train body and asks for the volume: volumes simply add, because the wood in one piece is still wood in the other. Q13 glues two blocks and asks for the area to be varnished, and areas do not simply add.

Truncated solids: sawing changes the faces (Q16)

A pen holder cut off on a slant is still made of pieces you can name — two trapezoid side faces, two rectangles, a base and a slanted top. Nothing new is needed. What changes is that the vertical edges are no longer all equal, so the trapezoid area formula does the work:

Here the two parallel sides are the two vertical edges, 10 cm and 7 cm, and the "height" of the trapezoid is the horizontal side it sits on. Sketch the face on its own, away from the solid, before deciding which length plays which role — inside a three-dimensional drawing the parallel sides look like heights and the height looks like a width.

Writing an area as an expression (Q5)

Q5 puts a letter where a number was: a prism with a square base of side x and a height of 9 cm. The method does not change. Six faces, two of them squares of area x² and four of them rectangles measuring x by 9, so

Part b) substitutes a value, and part c) asks you to redo the same computation face by face and compare. That is not busywork — it is the check that tells you whether the expression is right, and it is the only check available when there is a letter in the answer. Build the expression from the net, keep the squared term and the linear term apart, and do not collect x² with x.

Length, area, volume — and the units that give them away (Q11)

Q11 sorts five quantities and asks for a sensible unit for each. The unit is not decoration; it is the fastest way to catch an error, because a volume that comes out in cm² has been computed wrongly whatever the number says.

Faces, vertices, edges — and Euler's relation (Q7)

For a pyramid on an n-sided base: n + 1 faces, n + 1 vertices and 2n edges. For a prism on an n-sided base: n + 2 faces, 2n vertices and 3n edges. You can rebuild both counts from the net in a few seconds rather than memorising them, and Q7 then asks you to verify

on an octagonal pyramid. Q7 also asks which edges are congruent, which is worth a moment: the eight base edges are equal to each other and the eight sloping edges are equal to each other, but the two groups are not equal, and "regular" does not say they are.

The cube, and what happens when you scale it (Q17)

A cube of edge x has total area 6x² and volume x³. Q17 uses that in both directions: forwards from an edge of 7 cm, and backwards from a total area of 150 cm² — divide by 6 to get one face, then take the square root of that face to get the edge. Reversing the two steps, or forgetting the 6, is where this one goes wrong.

The scaling result is worth carrying into every question on the sheet. Multiply every edge by k and the area is multiplied by k², the volume by k³. Triple an edge and the area is nine times larger while the volume is twenty-seven times larger — the reason this feels surprising is that almost everyone expects both to triple.

Q18: the result that looks like a mistake

A rectangular bin with a cylindrical channel bored through it. The volume is the prism minus the cylinder, which is expected. The area is where it gets interesting: boring the hole removes two circular discs from the top and bottom faces, but it opens the whole curved wall of the channel, and that wall is larger than the two discs together. So the hole decreases the volume and increases the total area.

That is not a trick. It is the clearest demonstration on the sheet that area and volume are independent quantities measuring different things, and it is worth being able to explain in a sentence rather than only to compute.

Getting the most out of it

Draw the net before you write a formula

A rough net with every piece labelled turns an area question into an addition. It costs half a minute and it is the single habit that separates students who find this topic long from students who find it repetitive — which, once the net is automatic, it is.

Say which quantity you are computing, in units, on every line

Write cm, cm² or cm³ as you go, not only at the end. Most lost marks in this topic are a length used where an area belonged, and the unit catches that immediately while the number never does.

Do the naming questions as a review, at the end

The faces-vertices-edges questions and the polyhedron classification look like the easy ones and are usually done first. Come back to them after the area work instead: by then you have unfolded most of those solids, and the counting turns into a summary of everything you have just done.

Common questions